Economics » The Keynesian Perspective » The Expenditure-Output Model

# Building the Aggregate Expenditure Schedule

## Building the Aggregate Expenditure Schedule

Aggregate expenditure is the key to the expenditure-income model. The aggregate expenditure schedule shows, either in the form of a table or a graph, how aggregate expenditures in the economy rise as real GDP or national income rises. Thus, in thinking about the components of the aggregate expenditure line—consumption, investment, government spending, exports and imports—the key question is how expenditures in each category will adjust as national income rises.

### Consumption as a Function of National Income

How do consumption expenditures increase as national income rises? People can do two things with their income: consume it or save it (for the moment, let’s ignore the need to pay taxes with some of it). Each person who receives an additional dollar faces this choice. The marginal propensity to consume (MPC), is the share of the additional dollar of income a person decides to devote to consumption expenditures. The marginal propensity to save (MPS) is the share of the additional dollar a person decides to save. It must always hold true that:

$$\text{MPC + MPS = 1}$$

For example, if the marginal propensity to consume out of the marginal amount of income earned is 0.9, then the marginal propensity to save is 0.1.

With this relationship in mind, consider the relationship among income, consumption, and savings shown in this figure. (Note that we use “Aggregate Expenditure” on the vertical axis in this and the following figures, because all consumption expenditures are parts of aggregate expenditures.)

An assumption commonly made in this model is that even if income were zero, people would have to consume something. In this example, consumption would be $600 even if income were zero. Then, the MPC is 0.8 and the MPS is 0.2. Thus, when income increases by$1,000, consumption rises by $800 and savings rises by$200. At an income of $4,000, total consumption will be the$600 that would be consumed even without any income, plus $4,000 multiplied by the marginal propensity to consume of 0.8, or$ 3,200, for a total of $3,800. The total amount of consumption and saving must always add up to the total amount of income. (Exactly how a situation of zero income and negative savings would work in practice is not important, because even low-income societies are not literally at zero income, so the point is hypothetical.) This relationship between income and consumption, illustrated in this figure and this table, is called the consumption function. The Consumption Function In the expenditure-output model, how does consumption increase with the level of national income? Output on the horizontal axis is conceptually the same as national income, since the value of all final output that is produced and sold must be income to someone, somewhere in the economy. At a national income level of zero,$600 is consumed. Then, each time income rises by $1,000, consumption rises by$800, because in this example, the marginal propensity to consume is 0.8.

The pattern of consumption shown in this table is plotted in this figure. To calculate consumption, multiply the income level by 0.8, for the marginal propensity to consume, and add $600, for the amount that would be consumed even if income was zero. Consumption plus savings must be equal to income. The Consumption Function IncomeConsumptionSavings$0$600–$600
$1,000$1,400–$400$2,000$2,200–$200
$3,000$3,000$0$4,000$3,800$200
$5,000$4,600$400$6,000$5,400$600
$7,000$6,200$800$8,000$7,000$1,000
$9,000$7,800$1,200 However, a number of factors other than income can also cause the entire consumption function to shift. These factors were summarized in the earlier discussion of consumption, and listed in this table. When the consumption function moves, it can shift in two ways: either the entire consumption function can move up or down in a parallel manner, or the slope of the consumption function can shift so that it becomes steeper or flatter. For example, if a tax cut leads consumers to spend more, but does not affect their marginal propensity to consume, it would cause an upward shift to a new consumption function that is parallel to the original one. However, a change in household preferences for saving that reduced the marginal propensity to save would cause the slope of the consumption function to become steeper: that is, if the savings rate is lower, then every increase in income leads to a larger rise in consumption. ### Investment as a Function of National Income Investment decisions are forward-looking, based on expected rates of return. Precisely because investment decisions depend primarily on perceptions about future economic conditions, they do not depend primarily on the level of GDP in the current year. Thus, on a Keynesian cross diagram, the investment function can be drawn as a horizontal line, at a fixed level of expenditure. This figure shows an investment function where the level of investment is, for the sake of concreteness, set at the specific level of 500. Just as a consumption function shows the relationship between consumption levels and real GDP (or national income), the investment function shows the relationship between investment levels and real GDP. The Investment Function The investment function is drawn as a flat line because investment is based on interest rates and expectations about the future, and so it does not change with the level of current national income. In this example, investment expenditures are at a level of 500. However, changes in factors like technological opportunities, expectations about near-term economic growth, and interest rates would all cause the investment function to shift up or down. The appearance of the investment function as a horizontal line does not mean that the level of investment never moves. It means only that in the context of this two-dimensional diagram, the level of investment on the vertical aggregate expenditure axis does not vary according to the current level of real GDP on the horizontal axis. However, all the other factors that vary investment—new technological opportunities, expectations about near-term economic growth, interest rates, the price of key inputs, and tax incentives for investment—can cause the horizontal investment function to shift up or down. ### Government Spending and Taxes as a Function of National Income In the Keynesian cross diagram, government spending appears as a horizontal line, as in this figure, where government spending is set at a level of 1,300. As in the case of investment spending, this horizontal line does not mean that government spending is unchanging. It means only that government spending changes when Congress decides on a change in the budget, rather than shifting in a predictable way with the current size of the real GDP shown on the horizontal axis. The Government Spending Function The level of government spending is determined by political factors, not by the level of real GDP in a given year. Thus, government spending is drawn as a horizontal line. In this example, government spending is at a level of 1,300. Congressional decisions to increase government spending will cause this horizontal line to shift up, while decisions to reduce spending would cause it to shift down. The situation of taxes is different because taxes often rise or fall with the volume of economic activity. For example, income taxes are based on the level of income earned and sales taxes are based on the amount of sales made, and both income and sales tend to be higher when the economy is growing and lower when the economy is in a recession. For the purposes of constructing the basic Keynesian cross diagram, it is helpful to view taxes as a proportionate share of GDP. In the United States, for example, taking federal, state, and local taxes together, government typically collects about 30–35 % of income as taxes. This table revises the earlier table on the consumption function so that it takes taxes into account. The first column shows national income. The second column calculates taxes, which in this example are set at a rate of 30%, or 0.3. The third column shows after-tax income; that is, total income minus taxes. The fourth column then calculates consumption in the same manner as before: multiply after-tax income by 0.8, representing the marginal propensity to consume, and then add$600, for the amount that would be consumed even if income was zero. When taxes are included, the marginal propensity to consume is reduced by the amount of the tax rate, so each additional dollar of income results in a smaller increase in consumption than before taxes. For this reason, the consumption function, with taxes included, is flatter than the consumption function without taxes, as this figure shows.

The Consumption Function Before and After Taxes

The upper line repeats the consumption function from this figure. The lower line shows the consumption function if taxes must first be paid on income, and then consumption is based on after-tax income.

The Consumption Function Before and After Taxes

IncomeTaxesAfter-Tax IncomeConsumptionSavings
$0$0$0$600–$600$1,000$300$700$1,160–$460
$2,000$600$1,400$1,720–$320$3,000$900$2,100$2,280–$180
$4,000$1,200$2,800$2,840–$40$5,000$1,500$3,500$3,400$100
$6,000$1,800$4,200$3,960$240$7,000$2,100$4,900$4,520$380
$8,000$2,400$5,600$5,080$520$9,000$2,700$6,300$5,640$660

The export function, which shows how exports change with the level of a country’s own real GDP, is drawn as a horizontal line, as in the example in figure (a) where exports are drawn at a level of $840. Again, as in the case of investment spending and government spending, drawing the export function as horizontal does not imply that exports never change. It just means that they do not change because of what is on the horizontal axis—that is, a country’s own level of domestic production—and instead are shaped by the level of aggregate demand in other countries. More demand for exports from other countries would cause the export function to shift up; less demand for exports from other countries would cause it to shift down. The Export and Import Functions (a) The export function is drawn as a horizontal line because exports are determined by the buying power of other countries and thus do not change with the size of the domestic economy. In this example, exports are set at 840. However, exports can shift up or down, depending on buying patterns in other countries. (b) The import function is drawn in negative territory because expenditures on imported products are a subtraction from expenditures in the domestic economy. In this example, the marginal propensity to import is 0.1, so imports are calculated by multiplying the level of income by –0.1. Imports are drawn in the Keynesian cross diagram as a downward-sloping line, with the downward slope determined by the marginal propensity to import (MPI), out of national income. In figure (b), the marginal propensity to import is 0.1. Thus, if real GDP is$5,000, imports are $500; if national income is$6,000, imports are $600, and so on. The import function is drawn as downward sloping and negative, because it represents a subtraction from the aggregate expenditures in the domestic economy. A change in the marginal propensity to import, perhaps as a result of changes in preferences, would alter the slope of the import function. ### Using an Algebraic Approach to the Expenditure-Output Model In the expenditure-output or Keynesian cross model, the equilibrium occurs where the aggregate expenditure line (AE line) crosses the 45-degree line. Given algebraic equations for two lines, the point where they cross can be readily calculated. Imagine an economy with the following characteristics. Y = Real GDP or national income T = Taxes = 0.3Y C = Consumption = 140 + 0.9(Y – T) I = Investment = 400 G = Government spending = 800 X = Exports = 600 M = Imports = 0.15Y Step 1. Determine the aggregate expenditure function. In this case, it is: $$\begin{array}{rcl}\text{AE}& \text{=}& \text{C + I + G + X – M}\\ \text{AE}& \text{=}& \text{140 + 0.9(Y – T) + 400 + 800 + 600 – 0.15Y}\end{array}$$ Step 2. The equation for the 45-degree line is the set of points where GDP or national income on the horizontal axis is equal to aggregate expenditure on the vertical axis. Thus, the equation for the 45-degree line is: AE = Y. Step 3. The next step is to solve these two equations for Y (or AE, since they will be equal to each other). Substitute Y for AE: $$\begin{array}{rcl}\text{Y}& \text{=}& \text{140 + 0.9(Y – T) + 400 + 800 + 600 – 0.15Y}\end{array}$$ Step 4. Insert the term 0.3Y for the tax rate T. This produces an equation with only one variable, Y. Step 5. Work through the algebra and solve for Y. $$\begin{array}{rcl}\text{Y}& \text{=}& \text{140 + 0.9(Y – 0.3Y) + 400 + 800 + 600 – 0.15Y}\\ \text{Y}& \text{=}& \text{140 + 0.9Y – 0.27Y + 1800 – 0.15Y}\\ \text{Y}& \text{=}& \text{1940 + 0.48Y}\\ \text{Y – 0.48Y}& \text{=}& \text{1940}\\ \text{0.52Y}& \text{=}& \text{1940}\\ \cfrac{\text{0.52Y}}{\text{0.52}}& \text{=}& \cfrac{\text{1940}}{\text{0.52}}\\ \text{Y}& \text{=}& \text{3730}\end{array}$$ This algebraic framework is flexible and useful in predicting how economic events and policy actions will affect real GDP. Step 6. Say, for example, that because of changes in the relative prices of domestic and foreign goods, the marginal propensity to import falls to 0.1. Calculate the equilibrium output when the marginal propensity to import is changed to 0.1. $$\begin{array}{rcl}\text{Y}& \text{=}& \text{140 + 0.9(Y – 0.3Y) + 400 + 800 + 600 – 0.1Y}\\ \text{Y}& \text{=}& \text{1940 – 0.53Y}\\ \text{0.47Y}& \text{=}& \text{1940}\\ \text{Y}& \text{=}& \text{4127}\end{array}$$ Step 7. Because of a surge of business confidence, investment rises to 500. Calculate the equilibrium output. $$\begin{array}{rcl}\text{Y}& \text{=}& \text{140 + 0.9(Y – 0.3Y) + 500 + 800 + 600 – 0.15Y}\\ \text{Y}& \text{=}& \text{2040 + 0.48Y}\\ \text{Y – 0.48Y}& \text{=}& \text{2040}\\ \text{0.52Y}& \text{=}& \text{2040}\\ \text{Y}& \text{=}& \text{3923}\end{array}$$ For issues of policy, the key questions would be how to adjust government spending levels or tax rates so that the equilibrium level of output is the full employment level. In this case, let the economic parameters be: Y = National income T = Taxes = 0.3Y C = Consumption = 200 + 0.9(Y – T) I = Investment = 600 G = Government spending = 1,000 X = Exports = 600 Y = Imports = 0.1(Y – T) Step 8. Calculate the equilibrium for this economy (remember Y = AE). $$\begin{array}{rcl}\text{Y}& \text{=}& \text{200 + 0.9(Y – 0.3Y) + 600 + 1000 + 600 – 0.1(Y – 0.3Y)}\\ \text{Y – 0.63Y + 0.07Y}& \text{=}& \text{2400}\\ \text{0.44Y}& \text{=}& \text{2400}\\ \text{Y}& \text{=}& \text{5454}\end{array}$$ Step 9. Assume that the full employment level of output is 6,000. What level of government spending would be necessary to reach that level? To answer this question, plug in 6,000 as equal to Y, but leave G as a variable, and solve for G. Thus: $$\text{6000 = 200 + 0.9(6000 – 0.3(6000)) + 600 + G + 600 – 0.1(6000 – 0.3(6000))}$$ Step 10. Solve this problem arithmetically. The answer is: G = 1,240. In other words, increasing government spending by 240, from its original level of 1,000, to 1,240, would raise output to the full employment level of GDP. Indeed, the question of how much to increase government spending so that equilibrium output will rise from 5,454 to 6,000 can be answered without working through the algebra, just by using the multiplier formula. The multiplier equation in this case is: $$\begin{array}{rcl}\cfrac{\text{1}}{\text{1 – 0.56}}& \text{=}& \text{2.27}\end{array}\begin{array}{}\end{array}\phantom{\rule{0ex}{0ex}}$$ Thus, to raise output by 546 would require an increase in government spending of 546/2.27=240, which is the same as the answer derived from the algebraic calculation. This algebraic framework is highly flexible. For example, taxes can be treated as a total set by political considerations (like government spending) and not dependent on national income. Imports might be based on before-tax income, not after-tax income. For certain purposes, it may be helpful to analyze the economy without exports and imports. A more complicated approach could divide up consumption, investment, government, exports and imports into smaller categories, or to build in some variability in the rates of taxes, savings, and imports. A wise economist will shape the model to fit the specific question under investigation. ### Building the Combined Aggregate Expenditure Function All the components of aggregate demand—consumption, investment, government spending, and the trade balance—are now in place to build the Keynesian cross diagram. This figure builds up an aggregate expenditure function, based on the numerical illustrations of C, I, G, X, and M that have been used throughout this text. The first three columns in this table are lifted from the earlier this table, which showed how to bring taxes into the consumption function. The first column is real GDP or national income, which is what appears on the horizontal axis of the income-expenditure diagram. The second column calculates after-tax income, based on the assumption, in this case, that 30% of real GDP is collected in taxes. The third column is based on an MPC of 0.8, so that as after-tax income rises by$700 from one row to the next, consumption rises by $560 (700 × 0.8) from one row to the next. Investment, government spending, and exports do not change with the level of current national income. In the previous discussion, investment was$500, government spending was $1,300, and exports were$840, for a total of $2,640. This total is shown in the fourth column. Imports are 0.1 of real GDP in this example, and the level of imports is calculated in the fifth column. The final column, aggregate expenditures, sums up C + I + G + X – M. This aggregate expenditure line is illustrated in this figure. A Keynesian Cross Diagram Each combination of national income and aggregate expenditure (after-tax consumption, government spending, investment, exports, and imports) is graphed. The equilibrium occurs where aggregate expenditure is equal to national income; this occurs where the aggregate expenditure schedule crosses the 45-degree line, at a real GDP of$6,000. Potential GDP in this example is $7,000, so the equilibrium is occurring at a level of output or real GDP below the potential GDP level. ### National Income-Aggregate Expenditure Equilibrium National IncomeAfter-Tax IncomeConsumptionGovernment Spending + Investment + ExportsImportsAggregate Expenditure$3,000$2,100$2,280$2,640$300$4,620$4,000$2,800$2,840$2,640$400$5,080$5,000$3,500$3,400$2,640$500$5,540$6,000$4,200$3,960$2,640$600$6,000$7,000$4,900$4,520$2,640$700$6,460$8,000$5,600$5,080$2,640$800$6,920$9,000$6,300$5,640$2,640$900\$7,380

The aggregate expenditure function is formed by stacking on top of each other the consumption function (after taxes), the investment function, the government spending function, the export function, and the import function. The point at which the aggregate expenditure function intersects the vertical axis will be determined by the levels of investment, government, and export expenditures—which do not vary with national income. The upward slope of the aggregate expenditure function will be determined by the marginal propensity to save, the tax rate, and the marginal propensity to import. A higher marginal propensity to save, a higher tax rate, and a higher marginal propensity to import will all make the slope of the aggregate expenditure function flatter—because out of any extra income, more is going to savings or taxes or imports and less to spending on domestic goods and services.

The equilibrium occurs where national income is equal to aggregate expenditure, which is shown on the graph as the point where the aggregate expenditure schedule crosses the 45-degree line. In this example, the equilibrium occurs at 6,000. This equilibrium can also be read off the table under the figure; it is the level of national income where aggregate expenditure is equal to national income.

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